Jkbose maths syllabus

# Unit VI: Statistics and Probability

• ### Probability

Sets: Sets and their representations. Empty set. Finite and infinite sets. Equal sets, Subsets. Subsets of the set of real numbers especially intervals (with notations). Power set. Universal set. Venn diagrams, Union and Intersection of sets. Difference of sets. Complement of a set.

Relations and Functions: Relations and Functions Ordered pairs, Cartesian product of sets. Number of elements in the Cartesian product of two finite sets. Cartesian product of the reals with itself (up to RxRxR). Definition of relation, pictorial diagrams, domain, co-domain and range of relation. Function as a special kind of relation from one set to another. Pictorial representation of a function, domain and co-domain and range of a function – Real valued function of the real variable – domain and range of these functions. Constant, identity, polynomial, rational, modulus, signum and greatest integer functions with their graphs. Sum, difference product and quotients of functions.

Trigonometric Functions: Positive and negative angles. Measuring angles in radians and in degrees and conversion from one measure to another. Definition of trigonometric functions with the help of unit circle. Truth of the identity sin²x+cos²x = 1, for all x. Signs of trigonometric functions and sketch of their graphs. Expressing sin (x+y)and cos (x+y) in terms of sin x, sin y, cos x and cos y. Deducing the following identities.

Principle of Mathematical Induction :The Principle of Mathematical induction and Simple applications.

Permutation and Combinations :Fundamental principle of counting. Facturial n, Permutations and combinations, derivation of formulae and their connections, simple applications.

Complex Numbers and Linear Inequalities :Need for complex numbers, especially iota ( i ) to be motivated by inability to solve every quadratic equation. Brief description of algebraic properties of complex numbers. Argand plane and polar representation of complex numbers. Statement of Fundamental Theorem of Algebra, solution of quadratic equation in the complex number system. Linear inequalities: Algebraic solution of linear inequalities in one variable and their representation on the number line. Graphical solution of linear inequalities in two variable. Solution of system of linear inequalities in two variables- graphically.

Limits and Derivates:  Derivative introduced as rate of change both as that of distance function and geometrically, intuitive idea of limit. Definition of derivative, relate it to slope of tangent of the curve, derivative of sum, difference, product and quotient of functions. Derivatives of polynomial and trigonometric functions.

Straight Lines Brief recall of 2D from earlier classes: Slope of a line and angle between two lines. Various forms of equations of a line, parallel to axes, point – slope form, slope – intercept form, two – point form, intercepts form and normal form. General equation of a line. Distance of a point from a line.

Conic Sections Sections of a cone: Circles, Ellipse, Parabola, Hyperbola, a point, a straight line and a pair of intersecting lines as a degenerated case of a conic section. Standard equations and simple properties of Parabola, Ellipse and hyperbola, standard equation of a circle.

Probability Random experiments: Outcomes, Simple spaces (set representation). Events: Occurrence of events ‘not,’ ‘and’ ‘or’ events, mutually exclusive events. Axiomatic (set theoretic) probability, connections with the theories of earlier classes. Probability of an event, probability of ‘not’, ‘and’ & ‘or’ events.

Statistics Measure of dispersion: mean deviation, variance and standard deviation of ungrouped/ grouped data. Analysis of frequency distributions with equal means but different variances.

Binomial Theorem History, Statement and proof of the binomial theorem for positive integral indices Pascal’s triangle, general and middle term in binomial expansion simple applications.

Sequence and Series Sequence and Series. Arithmetic Progression (A.P.), arithmetic mean (A.M). Geometric progression (G.P) general term of a G.P sum of n terms if a G.P. Geometric mean (G.M), relation between A.M. and G.M. Sum to n terms of the special series :

Introduction to Three Dimensional Geometry Coordinates axes and coordinate planes in three dimensions. Coordinates of a point Distance between two points and section formula.

Mathematical Reasoning Mathematically acceptable statements. Connecting words/ phrases- consolidating the understanding of “if and only if (necessary and sufficient) conditions”, “implies,” “and/or”, “Implied by,” “and,” “or”, “there exists” and their use through variety of examples related to real life and Mathematics. Validating the statements involving the connecting words – difference between contradiction, converse and contrapositive.